Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}4x+y &= 1 \\ 5x-7y &= 2\end{align*}$
Explanation: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-5$ and the bottom equation by $4$ $\begin{align*}-20x-5y &= -5\\ 20x-28y &= 8\end{align*}$ Add the top and bottom equations. $-33y = 3$ Divide both sides by $-33$ and reduce as necessary. $y = -\dfrac{1}{11}$ Substitute $-\dfrac{1}{11}$ for $y$ in the top equation. $4x- \dfrac{1}{11} = 1$ $4x-\dfrac{1}{11} = 1$ $4x = \dfrac{12}{11}$ $x = \dfrac{3}{11}$ The solution is $\enspace x = \dfrac{3}{11}, \enspace y = -\dfrac{1}{11}$.